How to add local attributes of an operator to a `bpy.app.handle`?
This can be done by defining a function in the namespace where your variables are accessible:

    def invoke(self, context, event):
        self.param = {self: 'my dict'}

        print(bpy.app.handlers.frame_change_post)
        print(self.handle in bpy.app.handlers.frame_change_post) # Always False
        #bpy.app.handlers.frame_change_post.clear()

        # local function that can access this namespace
        def handle_local(scene):
            # pass any vars you want
            self.handle(self, scene, context)

        bpy.app.handlers.frame_change_post.append(handle_local)

        return {'RUNNING_MODAL'}


Note: the line print(self.handle in bpy.app.handlers.frame_change_post) is no longer correct, using a function defined each time makes this check more involved.